## Conditional Probability When the Sum of Two Geometric Random Variables Are Known

## Problem 755

Let $X$ and $Y$ be geometric random variables with parameter $p$, with $0 \leq p \leq 1$. Assume that $X$ and $Y$ are independent.

Let $n$ be an integer greater than $1$. Let $k$ be a natural number with $k\leq n$. Then prove the formula

\[P(X=k \mid X + Y = n) = \frac{1}{n-1}.\]